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Homotopy invariance of non-stable K1-functors

Published online by Cambridge University Press:  10 October 2013

A. Stavrova
A. Stavrova*
Affiliation:
Fields Institute for Research in Mathematical Sciences, Toronto, Canada and Department of Mathematics and Mechanics, St. Petersburg State University, Russia, [email protected]
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Abstract

Let G be a reductive algebraic group over a field k, such that every semisimple normal subgroup of G has isotropic rank ≥ 2, i.e. contains (Gm)2. Let K1G be the non-stable K1-functor associated to G, also called the Whitehead group of G. We show that K1G(k) = K1G (k[X1 ,…, Xn]) for any n ≥ 1. If k is perfect, this implies that K1G (R) = K1G (R[X]) for any regular k-algebra R. If k is infinite perfect, one also deduces that K1G (R) → K1G (K) is injective for any local regular k-algebra R with the fraction field K.

Keywords

Non-stable K1-functorWhitehead groupreductive groupelementary subgroupChevalley commutator formula19B2820G0720G1514L35
Type
Research Article
Information
Journal of K-Theory , Volume 13 , Issue 2 , April 2014 , pp. 199 - 248
Copyright
Copyright © ISOPP 2013 

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